Realization of quasicrystalline quadrupole topological insulators in electrical circuits

Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical circuits, and observe the spectrally and spatially localized corner modes. In measurement, the modes appear as topological boundary resonances in the corner impedance spectra. Additionally, we demonstrate the robustness of corner modes on the circuit. Our circuit design may be extended to study topological phases in higher-dimensional aperiodic structures.

The understanding of topological phases of matter has always been based on the topological band theory, which is defined in crystalline materials with long-range order and periodicity. To the best of our knowledge, topological phases are observed only in one-dimensional quasicrystals [26]. Although there are predictions of topological phases in 2D quasicrystals [27][28][29][30][31][32][33][34][35][36][37][38], the experimental observations have never been reported.
Using impedance measurement, the corner states protected by quantized quadrupole moment are directly observed in the circuits. The robustness of topological corner states is also demonstrated. Our work provides the experimental evidence of topological phases in 2D quasicrystalline systems. We expect more topological states can be observed in circuits based on similar implementations.

Quasicrystalline quadrupole topological insulators
We construct the QTI on an AB tiling quasicrystal by only considering nearestand next-nearest-neighbor hoppings [34]. The tight-binding Hamiltonian is ( ) and jk φ indicates the polar angle of bond between site j and k with respect to the horizontal direction. Here is the creation operator in cell j.
The first and second terms are the intracell and intercell hoppings with amplitudes A0 and A1, respectively. 4 and R is an orthogonal matrix permuting the sites of the tiling to rotate the whole system by / 4 π .
The rotational symmetry 4 results in a quantized quadrupole moment 0, / 2 xy Q e = . Thus, xy Q is a natural topological invariant, which can be calculated in real space [53,54]. By numerical calculations, we confirmed that the AB tiling quasicrystal has the non-trivial quadrupole moment

Realization in electrical circuits
The quasicrystalline lattice can be mapped to an electrical circuit. To realize quasicrystalline QTI, we design an electrical circuit depicted in Figure. 1a. The circuit is characterized by the Kirchhoff's law where in cell p(q), and ω is the frequency of the circuit. The circuit Laplacian is with , pa qb C being the capacitance between two sites, and As shown in Figure. 1a, the solid and dashed lines correspond to hoppings with positive and negative values, which can be realized by the capacitors and inductors, respectively. We choose capacitor C1 and inductor L1 for intracell hopping, and C2 and L2 for intercell hopping. The nearest neighbor couplings between cells are introduced to stabilize the corner states in small size systems [34]. If these couplings is negleted, the exact AB tiling is restored, and the physical results don't change qualitatively.
Under a suitable choice of the grounded elements, the tight-binding Hamiltonian of the QTI on a quasicrystalline lattice in Eq. 1 is mapped to the circuit Hamiltonian in Eq. 2. Note that there exists a relation C2 / C1=L1 / L2=A1 /A0.
The two-point impedance is: where Va(b) is the voltage on site a(b), Iab is the current between the two sites, and jn and n ψ are the eigenvalue and eigenstates of the matrix ( ) J ω , respectively [21].
The two-point impedanace Zab diverges in the presence of zero-admittance modes (jn = 0) with , , n a n b ψ ψ ≠ . Hence, the corner states with zero-admittance can be observed by measuring the two-point impedance.
We consider the QTI on an AB tiling 2D quasicrystalline lattice. A circuit that realizes the QTIs is depicted in Figure.  To realize the QTI experimentally, a circuit with 69 unit cells was fabricated as shown in Figure. 1b. The intracell elements with C1 = 10 nF and L1 = 1 mH result in a resonant frequency 50.3 kHz (parameters for other elements are given in Methods).
Here we set the coupling ratio 10 λ = in order to get highly localized corner states, which can be observed by two-point impedance measurements between the corner/edge/bulk site and another bulk site using an impedance analyzer.  Figure. 3b, the corner states are spectrally localized with a deviation from ω0. Figures. 3c-f show the spatial impedance distributions at the frequencies of the four corner states. Compared with the spatial impedance distribution in Figure. 2b, although the maximum impedances are located at only one of the four corner sites, the topological corner states are still spatially localized. The spectral and spatial localizations imply the robustness of corner states on the circuit.

Discussion
The quadruple insulators were always realized on periodic systems with translational symmetry, and the quadruple moments are well defined in momentum space. Unlike the periodic structures, the quasi-periodic systems possess long-range order but don't have translational symmetry. Therefore, quasicrystalline insulators with a quantized quadrupole moment extend the concept of quadruple insulators defined in crystals. The realization of quadruple isolator in this work opens a way for the implementation of corner states in more quasi-periodic systems.
The circuit implementation of the 2D quasicrystalline QTIs confirms the existence and robustness of the corner states on the circuit. This work provides an experimental evidence for the first time to implement the topological phase in a 2D quasicrystalline system, and extends the territory of topological phases beyond crystals to higher-dimensional aperiodic systems. The highly customizable circuit platform can also readily be applied to other 2D quasicrystals with different symmetries and to 3D quasicrystalline structures [55], which may realize more exotic topological states. Here we set the coupling ratio λ = 10 in order to get highly localized corner states, and the two-point impedance measurement is carried out using an Impedance Analyzer 4192A LF.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
microwave quadrupole insulator with topologically protected corner states. Nature.